Remark 3.6  However, extremal property (REF ) cannot serve as an alternative characterization of inner balayage, for it does not determine \(\mu ^A\)  uniquely within \(\Gamma _{A,\mu }^+\) . Indeed, consider the \(\alpha \) -Riesz kernel \(|x-y|^{\alpha -n}\)  of order \(\alpha \leqslant 2\) , \(\alpha <n\) , on \(\mathbb {R}^n\) , \(n\geqslant 2\) , and a proper, closed subset \(A\)  of \(\mathbb {R}^n\)  that is not \(\alpha \) -thin at infinity [1]} (take, for instance, \(A:=\lbrace |x|\geqslant 1\rbrace \) ). Then for any \(\mu \in \mathcal {E}^+_{A^c}\) ,
\(\mu ^A\ne \mu \text{ \ and \ }\mu ^A(\mathbb {R}^n)=\mu (\mathbb {R}^n),\) 
